Multiplicative mappings at some points on matrix algebras

Let M_n be the algebra of all n×n matrices, and let φ:M_n→M_n be a linear mapping. We say that φ is a multiplicative mapping at G if φ(ST)=φ(S)φ(T) for any S,T∈M_n with ST=G. Fix G∈M_n, we say that G is an all-multiplicative point if every multiplicative linear bijection φ at G with φ(I_n)=I_n is a multiplicative mapping in M_n, where I_n is the unit matrix in M_n. We mainly show in this paper the following two results: (1) If G∈M_n with detG=0, then G is an all-multiplicative point in M_n; (2) If φ is an multiplicative mapping at I_n, then there exists an invertible matrix P∈M_n such that either φ(S)=PSP^-1 for any S∈M_n or φ(T)=PT^trP^-1 for any T∈M_n.     

ADDITIVE FAMILIES OF INVARIANTS OF SKEWSYMMETRIC TENSORS

Abstract

Let d be a positive integer and F be a field of characteristic 0. Suppose that for each positive integer n, I _n is a polynomial invariant of the usual action of GL_n (F) on Λ^d(Fⁿ), such that for t ? Λ^d(F ^k) and s ? Λ^d(F ^l), I _{k + l} (t l s) = I _k(t)I _t (s), where t ⊥ s is defined in §1.4. Then we say that {I_n} is an additive family of invariants of the skewsymmetric tensors of degree d, or, briefly, an additive family of invariants. If not all the I_n are constant we say that the family is non-trivial. We show that in each even degree d there is a non-trivial additive family of invariants, but that this is not so for any odd d. These results are analogous to those in our paper [3] for symmetric tensors. Our proofs rely on the symbolic method for representing invariants of skewsymmetric tensors. To keep this paper self-contained we expound some of that theory, but for the proofs we refer to the book [2] of Grosshans, Rota and Stein.     

A Hadamard product involving N-matrices

We show that for any pair M,N of n by n M-matrices, the Hadamard (entry-wise) product M°N _-1 is again an M-matrix. For a single M-matrix M, the matrix M°M ^-1 is also considered.     

Proof of a conjecture of José L. Rubio de Francia

Given a compact connected abelian group G, its dual group Γ can be ordered (in a non-canonical way) so that it becomes an ordered group. It is known that, for any such ordering on Γ and p in the range 1<p<∞, the characteristic function χ_I of an interval I in Γ is a p—multiplier with a uniform bound (independent of I) on the corresponding operator S_I on L^p(G). In this note it is shown that, for 1<p,q<∞, there is a constant C_p,q, independent of G and the particular ordering on Γ, such that for all sequences {I_j} of intervals in Γ and all sequences {f_j} in L^p(G). Such a result was conjectured by J.L. Rubio de Francia, who noted its validity when The present proof uses a transference argument, an approach which shows that any constant C_p,q for which the inequality holds when G = will serve for every G and every ordering on Γ. An added advantage of this approach is that it adapts to give an extension of the result for functions taking values in a UMD space.The work of the first author was partially supported by a grant from the National Science Foundation (U.S.A.). The second and third authors were partially supported by the HARP network HPRN-CT-2001-00273 of the European Commission and by grant BFM2001-0188 of Ministerio de Ciencia y Tecnologia.     

Finitely generated varieties of distributive double <Emphasis Type="Italic">p</Emphasis>-algebras universal modulo a group

For any monoid M, any universal variety contains arbitrarily large algebras whose endomorphism monoid is isomorphic to M. A variety universal modulo a group G contains arbitrarily large algebras whose endomorphism monoid is isomorphic to the direct product M x G. One of the results of this paper structurally characterizes all finitely generated varieties of distributive double p-algebras universal modulo a group, and shows that any unavoidable direct factor G is a Boolean group with at most eight elements.     

The inertia set of the join of graphs

Let G=(V,E) be a graph with V={1,2,…,n}. Denote by S(G) the set of all real symmetric n×n matrices A=[a_i,j] with a_i,j≠0, i≠j if and only if ij is an edge of G. Denote by I^↗(G) the set of all pairs (p,q) of natural numbers such that there exists a matrix A∈S(G) with at most p positive and q negative eigenvalues. We show that if G is the join of G₁ and G₂, then I^↗(G)?{(1,1)}=I^↗(G₁∨K₁)∩I^↗(G₂∨K₁)?{(1,1)}. Further, we show that if G is a graph with s isolated vertices, then , where denotes the graph obtained from G be removing all isolated vertices, and we give a combinatorial characterization of graphs G with (1,1)∈I^↗(G). We use these results to determine I^↗(G) for every complete multipartite graph G.     

Strongly self-dual graphs

We present a class of graphs whose adjacency matrices are nonsingular with integral inverses, denoted h-graphs. If the h-graphs G and H with adjacency matrices M(G) and M(H) satisfy M(G)^-1=SM(H)S, where S is a signature matrix, we refer to H as the dual of G. The dual is a type of graph inverse. If the h-graph G is isomorphic to its dual via a particular isomorphism, we refer to G as strongly self-dual. We investigate the structural and spectral properties of strongly self-dual graphs, with a particular emphasis on identifying when such a graph has 1 as an eigenvalue.     

QUASI-FROBENIUS BIMODULES OF FUNCTIONS ON A SEMIGROUP

Let _AM_B be a QF-bimodule, A a left Artinian ring, B a right Artinian ring, G a semigroup with a unit element (a monoid). Let M^G be the set of all functions on G with values in M. Consider M^G as an (AG, BG)-bimodule over the semigroup rings AG and BG. It is proved that the annihilator maps I → r_M^G (I) and R → l_AG (R) are mutually inverse bijective Galois correspondences between the set of finitely cogenerated left ideals I ? AG and the set of right BG-submodules R ? M^G finitely generated over B. The maps J → l_M^G (J) and L → r_AG (L) are mutually inverse bijective Galois correspondences between the set of finitely cogenerated right ideals J ? AG and the set of left AG-submodules L ? M^G finitely generated over A. This result also makes it possible, starting from a given QF-bimodule _A M_B , to construct new QF-bimodules _AG/IS_BG/J as bimodules of functions on a semigroup with values in M.     

Circular Chromatic Number and Mycielski Graphs

As a natural generalization of graph coloring, Vince introduced the star chromatic number of a graph G and denoted it by ^*(G). Later, Zhu called it circular chromatic number and denoted it by _c(G). Let (G) be the chromatic number of G. In this paper, it is shown that if the complement of G is non-hamiltonian, then _c(G)=(G). Denote by M(G) the Mycielski graph of G. Recursively define M^m(G)=M(M^m–1(G)). It was conjectured that if mn–2, then _c(M^m(K_n))=(M^m(K_n)). Suppose that G is a graph on n vertices. We prove that if , then _c(M(G))=(M(G)). Let S be the set of vertices of degree n–1 in G. It is proved that if |S| 3, then _c(M(G))=(M(G)), and if |S| 5, then _c(M²(G))=(M²(G)), which implies the known results of Chang, Huang, and Zhu that if n3, _c(M(K_n))=(M(K_n)), and if n5, then _c(M²(K_n))=(M²(K_n)).* Research supported by Grants from National Science Foundation of China and Chinese Academy of Sciences.     

An affine characterization of the Veronese surface

IfM ² is a nondegenerate surface in a 4-dimensional Riemannian manifold , then there is a natural affine metricg defined onM ². It is shown that this affine metricg is conformal to the induced Riemannian metric onM ² if and only ifM ² is a minimal submanifold of in the usual Riemannian sense. If the conformal factor is a constant, then the two metrics are said to be homothetic. It is shown that there does not exist a nondegenerate surface in Euclidean space ⁴ or hyperbolic spaceH ⁴ whose affine metric is homothetic to the induced Riemannian metric. Furthermore, ifM ² is a nondegenerate surface in the standard 4-sphereS ⁴ whose affine metric is homothetic to the induced Riemannian metric, thenM ² is a Veronese surface.T. Cecil was supported by NSF Grant No. DMS-9101961.     

On simple modules and normal subgroups of p-solvable groups

Let N be a normal subgroup of a p-solvable group G and let M be a simple FN-module, where F is an algebraically closed field of characteristic p. Next, denote by IRR⁰(FG|M) the set of all simple FG-modules V lying over M such that the p-part of dim_F V is as small as possible. In this paper, |IRR⁰(FG|M)| and the vertices of modules in IRR⁰(FG|M) are determined. The p-blocks of G to which modules in IRR⁰(FG|M) belong are also determined.Received: 5 December 2003     

Isoperimetric inequalities in potential theory

Given a non-empty bounded domainG in ⁿ ,n2, letr ₀(G) denote the radius of the ballG ₀ having center 0 and the same volume asG. The exterior deficiencyd _e (G) is defined byd _e (G)=r _e (G)/r ₀(G)–1 wherer _e (G) denotes the circumradius ofG. Similarlyd _i (G)=1–r _i (G)/r ₀(G) wherer _i (G) is the inradius ofG. Various isoperimetric inequalities for the capacity and the first eigenvalue ofG are shown. The main results are of the form CapG(1+cf(d _e (G)))CapG ₀ and ₁(G)(1+cf(d _i (G)))₁(G ₀),f(t)=t ³ ifn=2,f(t)=t ³/(ln 1/t) ifn=3,f(t)=t ^(n+3)/2 ifn4 (for convex G and small deficiencies ifn3).     

Separation of representations with quadratic overgroups

Any unitary irreducible representation π of a Lie group G defines a moment set Iπ, subset of the dual g^? of the Lie algebra of G. Unfortunately, Iπ does not characterize π. If G is exponential, there exists an overgroup G⁺ of G, built using real-analytic functions on g^?, and extensions π⁺ of any generic representation π to G⁺ such that Iπ⁺ characterizes π.In this paper, we prove that, for many different classes of group G, G admits a quadratic overgroup: such an overgroup is built with the only use of linear and quadratic functions.     

Dominating sets of the comaximal and ideal-based zero-divisor graphs of commutative rings

Abstract

Let R be a commutative ring with nonzero identity, and let I be an ideal of R. The ideal-based zero-divisor graph of R, denoted by Γ_I (R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y∈ R \ I} and two distinct vertices x and y are adjacent if and only if xy ∈ I. Define the comaximal graph of R, denoted by CG(R), to be a graph whose vertices are the elements of R, where two distinct vertices a and b are adjacent if and only if Ra+Rb=R. A nonempty set S ? V of a graph G=(V, E) is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this paper is to study the dominating sets and domination number of Γ_I (R) and the comaximal graph CG₂(R) \ J (R) (or CGJ (R) for short) where CG₂(R) is the subgraph of CG(R) induced on the nonunit elements of R and J (R) is the Jacobson radical of R.     

A comparison of length definitions for maps of modules over local rings

LetM be a finitely generated free module over a local ring. An automorphism ofM can be written as a product of automorphisms that are elements of a given generating groupG of the set of all automorphisms ofM. The minimal number of elements required for such a product is called the length of. We study two decompositions using similar generating groups and compare the two resulting lengths.     

Maximum nullity of outerplanar graphs and the path cover number

Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[a_ij] with a_ij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).     

Solution to a problem of C. D. Godsil regarding bipartite graphs with unique perfect matching

We give the solution to the following question of C. D. Godsil[2]: Among the bipartite graphsG with a unique perfect matching and such that a bipartite graph obtains when the edges of the matching are contracted, characterize those having the property thatG ⁺G, whereG ⁺ is the bipartite multigraph whose adjacency matrix,B ⁺, is diagonally similar to the inverse of the adjacency matrix ofG put in lower-triangular form. The characterization is thatG must be obtainable from a bipartite graph by adding, to each vertex, a neighbor of degree one. Our approach relies on the association of a directed graph to each pair (G, M) of a bipartite graphG and a perfect matchingM ofG.     

Finite 2-groups with a nonabelian Frattini subgroup of order 16

According to a classical result of Burnside, if G is a finite 2-group, then the Frattini subgroup Φ(G) of G cannot be a nonabelian group of order 8. Here we study the next possible case, where G is a finite 2-group and Φ(G) is nonabelian of order 16. We show that in that case Φ(G) ≅ M × C₂, where M ≅ D₈ or M ≅ Q₈ and we shall classify all such groups G (Theorem A). Received: 16 February 2005; revised: 7 March 2005     

Lattices and infinite-dimensional forms. ‘The Lattice Method’

Hermitean vector spaces E of infinite dimensions are considered. Let G be a subgroup of the orthogonal group of E acting on a set M. The Lattice Method is a technique for classifying the orbits in M under G. We discuss the method in abstract terms and we illustrate it by means of three classification results showing that it is decisive to do a considerable amount of explicit calculations with vector subspace lattices.